Problem 23 - Beginning and Intermediate Algebra

Question

Evaluate each polynomial when \(x=-4\) and \(y=3\) $$2 x y-7 x+9$$

Step-by-Step Solution

Answer

When \(x=-4\) and \(y=3\), the polynomial \(2xy-7x+9\) evaluates to 13.

Step 1: Substitute given x and y values into the polynomial

Let's substitute x = -4 and y = 3 into the given polynomial: \(2xy - 7x + 9\).

Step 2: Perform operations

Now, we'll perform the operations on the substituted values.

Step 3: Solve the expression with the given x and y values

We have the expression: \(2(-4)(3) - 7(-4) + 9\), which we'll solve step by step. 1. First, solve for \(2(-4)(3)\): \[2(-4)(3) = -8(3) = -24\] 2. Next, solve for \(-7(-4)\): \[-7(-4) = 28\] 3. Finally, plug these values back into the expression and compute the result: \[-24 + 28 + 9 = 4 + 9 = 13\] So, the polynomial evaluates to 13 when x = -4 and y = 3.

Key Concepts

Substitution Method

The substitution method is a straightforward technique used to evaluate expressions like polynomials. Here, specific values are substituted for the variables within the expression. In our example, we are given the polynomial expression \(2xy - 7x + 9\) and asked to evaluate it for \(x = -4\) and \(y = 3\).

Understanding the substitution method involves knowing why we replace variables with specific values. It is useful for determining the numerical value of expressions when conditions or particular inputs are provided. This method ensures we transform algebraic expressions with variables into simple arithmetic problems, which are easy to calculate.

• Start by identifying the variables in the expression.
• Replace these variables with the given numerical values.
• Compute the arithmetic operations to find the final result.

This method is essential for evaluating polynomials and understanding how inputs affect outcomes in algebraic contexts.

Algebraic Expressions

Algebraic expressions form the building blocks of equations and are made up of variables and constants, combined using arithmetic operations. The expression \(2xy - 7x + 9\) is an example of how variables \(x\) and \(y\) interact with numbers to form a coherent mathematical statement.

Understanding algebraic expressions is crucial for solving many mathematical problems. Here are some elements you encounter in such expressions:

• Variables: Symbols like \(x\) and \(y\) that can represent numbers.
• Constants: Fixed values, like 2, -7, and 9 in our expression.
• Operators: Mathematical symbols indicating operations, such as addition (-) and multiplication (*).

By understanding these components and how they relate, you can interpret and manipulate expressions more effectively. This foundation prepares you for more complex operations and functions in algebra.

Polynomial Operations

Performing operations on polynomials involves arithmetic with algebraic expressions, like addition, subtraction, and multiplication. In the example \(2xy - 7x + 9\), operations must be completed in a specific order, respecting the hierarchy of calculations: multiplication before addition or subtraction.

To evaluate the polynomial after substitution, follow these steps:

1. **Multiply:** First, apply the multiplication to terms such as \(2(-4)(3)\), resulting in \(-24\).
2. **Multiply Again:** Next, tackle the term \(-7(-4)\), yielding \(28\).
3. **Add:** Finally, add the resulting values \(-24 + 28 + 9\) to arrive at the solution \(13\).

These operations are an extension of basic arithmetic but are applied with variables and coefficients, making them "polynomial operations." Skills in these calculations help solve real-world problems modeled by polynomial expressions and functions.